Five-Minute Check (over Chapter 9)CCSSThen/NowNew VocabularyKey Concept: Sequences as FunctionsExample 1: Identify Arithmetic SequencesExample 2: Graph an Arithmetic SequenceExample 3: Real-World Example: Find a TermExample 4: Identify Geometric SequencesExample 5: Graph a Geometric SequenceExample 6: Classify Sequences

Over Chapter 9

Find the midpoint of the line segment with endpoints at (–4, 9) and (5, –17).

A.

B.

C. (1, –5)

D.

Over Chapter 9

Find the distance between the points at (–5, 2) and (–9, 7).

A.

B. 6 units

C.

D.

Over Chapter 9

Graph x2 – 8x – y + 19 = 0.

A.

B.

C.

D.

Over Chapter 9

Graph 3x2 + 6x + y2 – 6y = –3.

A.

B.

C.

D.

Over Chapter 9

Which of the following equations represents a circle?

A. 4(x + 2)2 + 9(y – 3)2 = 36

B. x – 3 – (y – 2)2 = –2

C. (y – 2)2 – 9(x – 3)2 = 1

D. (x + 6)2 + y2 = 81

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Content StandardsF.IF.4 For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship.Mathematical Practices2 Reason abstractly and quantitatively.7 Look for and make use of structure.

You analyzed linear and exponential functions.

• Relate arithmetic sequences to linear functions.

• Relate geometric sequences to exponential functions.

• sequence

• term

• finite sequence

• infinite sequence

• arithmetic sequence

• common difference

• geometric sequence

• common ratio

Identify Arithmetic Sequences

A. Determine whether the sequence is arithmetic.–3, –8, –13, –23, …

There is no common difference.

Answer: This is not an arithmetic sequence.

–3 –8 –13 –23

–5 –5 –10

Identify Arithmetic Sequences

B. Determine whether the sequence is arithmetic.–8, –2, 4, 10, …

The common difference is 6.

Answer: The sequence is arithmetic.

–8 –2 4 10

+6 +6 +6

A. The common difference is 9.

B. The common difference is 11.

C. The common difference is 13.

D. The sequence is not arithmetic.

A. Determine whether the sequence is arithmetic. If so, determine the common difference.–16, –5, 6, 17, …

A. The common difference is –8.

B. The common difference is 5.

C. The common difference is 8.

D. The sequence is not arithmetic.

B. Determine whether the sequence is arithmetic. If so, determine the common difference.22, 14, 6, 2, …

Graph an Arithmetic Sequence

A. Consider the arithmetic sequence –8, –6, –4, ….Find the next four terms of the sequence.

Step 1 To determine the common difference, subtractany term from the term directly after it. Thecommon difference is –4 – (–6) or 2.

Step 2 To find the next term, add 2 to the last term.Continue to add 2 to find the following terms.

Answer: The next four terms are –2, 0, 2, and 4.

–8 –6 –4 –2 0 2 4

+2 +2+2 +2 +2+2

Graph an Arithmetic Sequence

B. Consider the arithmetic sequence –8, –6, –4, ….Graph the first seven terms of the sequence.

The domain contains the terms {1, 2, 3, 4, 5, 6, 7} and the range contains the terms {–8, –6, –4, –2, 0, 2, 4}. So, graph the corresponding pairs.

Answer:

A. 4, 8, 12, and 16

B. 6, 15, 24, and 33

C. 3, 12, 21, and 30

D. 5, 14, 23, and 32

A. Consider the arithmetic sequence 22, 13, 4, … . Find the next four terms of the sequence.

B. Consider the arithmetic sequence 22, 13, 4, … . Graph the first seven terms of the sequence.A. B.

C. D.

Find a Term

MARCHING BAND During their routine, a high school marching band marches in rows. There is one performer in the first row, three performers in the next row, and five in the third row. This pattern continues for the rest of the rows. Use this information to determine how many performers will be in the 20th row during the routine.Understand Because the difference between any

two consecutive rows is 2, the commondifference for the sequence is 2.

Plan Use point-slope form to write an equationfor the sequence. Let m = 2 and(x1, y1) = (3, 5). Then solve for x = 20.

Find a Term

Solve (y – y1) = m(x – x1) Point-slope form

(y – 5) = 2(x – 3) m = 2 and(x1, y1) = (3, 5)

y – 5 = 2x – 6 Multiply.y = 2x – 1 Add 5 to each

side.y = 2(20) – 1 Replace x

with 20.y = 40 – 1 or 39 Simplify.

Find a Term

Check You can find the terms of the sequence byadding 2, starting with row 1, until you reach 20.

Answer: There will be 39 performers in the 20th row.

A. 41 blocks

B. 45 blocks

C. 49 blocks

D. 53 blocks

PYRAMIDS Hermán is building a pyramid out of blocks for an engineering class. On the top level, there is one block. In the second level, there are 5 blocks. In the third, there are 9 blocks. This pattern continues for the rest of the levels down to the 18th level at the base of the pyramid. Use this information to determine how many blocks will be in the 13th level of the pyramid.

Identify Geometric Sequences

A. Determine whether the sequence is geometric.8, 20, 50, 125, …

Find the ratios of the consecutive terms.

Answer: The ratios are the same, so the sequence is geometric.

Identify Geometric Sequences

B. Determine whether the sequence is geometric.19, 30, 41, 52, …

Answer: The ratios are not the same, so the sequence is not geometric.

A. AB. B

A. The sequence is geometric.

B. The sequence is not geometric.

A. Determine whether the sequence is geometric.–4, 8, –16, 32, …

A. The sequence is geometric.

B. The sequence is not geometric.

B. Determine whether the sequence is geometric.2, 9, 40.5, 121.5, …

Graph a Geometric Sequence

A. Consider the geometric sequence 10, 15, 22.5, … .Find the next three terms of the sequence.

Step 2 To find the next term, multiply the previous

term by Continue multiplying by to find the

following terms.

Step 1 Find the value of the common ratio:

Graph a Geometric Sequence

Answer: The next three terms are 33.75, 50.625, and 75.938.

10 15 22.5 33.75 50.625 75.938

Graph a Geometric Sequence

B. Graph the first six terms of the sequence.

Domain: {1, 2, 3, 4, 5, and 6}Range: {10, 15, 22.5, 33.75, 50.625, 75.938}

Answer:

A. 30, 42, 54

B. 36, 72, 144

C. 72, 288, 1152

D. 54, 162, 486

A. Consider the geometric sequence 2, 6, 18, … . Find the next three terms of the sequence.

B. Graph the first six terms of the sequence.

A. B.

C. D.

Classify Sequences

A. Determine whether the sequence is arithmetic, geometric, or neither. Explain your reasoning.13, 25, 37, 49, …

Check for a common difference.49 – 37 = 12 37 – 25 = 12 25 – 13 = 12

Answer: Because there is a common difference, the sequence is arithmetic.

Check for a common ratio.

Classify Sequences

B. Determine whether the sequence is arithmetic, geometric, or neither. Explain your reasoning.2, 5, 9, 14, …

Check for a common difference.14 – 9 = 5 9 – 5 = 4

Answer: Because there is no common difference or common ratio, the sequence is neither arithmetic nor geometric.

Check for a common ratio.

Classify Sequences

C. Determine whether the sequence is arithmetic, geometric, or neither. Explain your reasoning.6, –12, 24, –48, …

Check for a common difference.–48 – 24 = –72 24 – (–12) = 36

Answer: Because there is a common ratio, the sequence is geometric.

Check for a common ratio.

A. The sequence is arithmetic.

B. The sequence is geometric.

C. The sequence is neither.

A. Determine whether the sequence is arithmetic, geometric, or neither. Explain your reasoning.8, 24, 48, 96, …

A. The sequence is arithmetic.

B. The sequence is geometric.

C. The sequence is neither.

B. Determine whether the sequence is arithmetic, geometric, or neither. Explain your reasoning.5, 12, 19, 26, …

A. The sequence is arithmetic.

B. The sequence is geometric.

C. The sequence is neither.

C. Determine whether the sequence is arithmetic, geometric, or neither. Explain your reasoning.300, 200, …